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arXiv:2505.02204 (Published 2025-05-04)
Semi-integral points of bounded height on vector group compactifications
Categories: math.NTIn this article, we obtain the asymptotic formula for the counting function of Darmon points of bounded height on equivariant compactifications of vector groups using ideas similar to those in [PSTVA21]. We also calculate the leading constants in some examples.
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arXiv:1801.05779 (Published 2018-01-17)
Rationality of Darmon points over genus fields of non maximal orders
Comments: 19 pagesCategories: math.NTStark--Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show that Darmon points of conductor $c\geq 1$ twisted by quadratic characters of $G_c^+=Gal(H_c^+/F)$, where $H_c^+$ is the strict ring class field of $F$ of conductor $c$, come from rational points on the elliptic curve defined over $H_c^+$.
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arXiv:1709.06929 (Published 2017-09-20)
An automorphic approach to Darmon points
Categories: math.NTWe give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a unifying point of view, emphasizing the automorphic nature of the construction.
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The Saito-Kurokawa lifting and Darmon points
Comments: 14 pages. Title changedCategories: math.NTKeywords: darmon points, fourier coefficients, siegel modular forms admits, positive definite symmetric half-integral matrices, adic saito-kurokawa liftTags: journal articleLet $E_{/_\Q}$ be an elliptic curve of conductor $Np$ with $p\nmid N$ and let $f$ be its associated newform of weight 2. Denote by $f_\infty$ the $p$-adic Hida family passing though $f$, and by $F_\infty$ its $\Lambda$-adic Saito-Kurokawa lift. The $p$-adic family $F_\infty$ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{\widetilde A_T(k)\}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$. We relate explicitly certain global points on $E$ (coming from the theory of Stark-Heegner points) with the values of these Fourier coefficients and of their $p$-adic derivatives, evaluated at weight $k=2$.